Question

What is m(angle)DFE

A)108

B)59

C)67

D)126

Answer 1

check the picture below.

so, by the intersecting chords theorem, those two vertical angles are 121°.

now, a circle has a total of 360°, so, if we take those two angles out, we're left with 360° - 121° - 121°, or 118°.

now, angle F and its twin, the vertical angle across from the junction, take up that remaining 118°, namely each one is 59°.

Answer 2

B.
`59^(\circ)`

Explanation:

We have been given an image of a circle and we are asked to find the measure of angle DFE.

We can see that angle DFE is formed by two intersecting chords inside the circle, so the measure of angle DFE will half the sum of its intercepted arcs (Arc BC+ Arc DE).

Since we know that the measure of circumference of a circle is 360 degrees, so we can set an equation to find the measure of arc BC and DE as:

`\text{Arc CB+Arc DE+Arc BD+Arc +CE}=360^(\circ)`

`\text{Arc CB+Arc DE}+108^(\circ)+134^(\circ)=360^(\circ)`

`\text{Arc CB+Arc DE}+242^(\circ)=360^(\circ)`

`\text{Arc CB+Arc DE}+242^(\circ)-242^(\circ)=360^(\circ)-242^(\circ)`

`\text{Arc CB+Arc DE}=118^(\circ)`

Now we will find the measure of angle DFE using intersecting secants theorem.

`m\angle DFE=(1)/(2)* \text{(Arc CB+Arc DE)}`

`m\angle DFE=(1)/(2)* 118^(\circ)`

`m\angle DFE=59^(\circ)`

Therefore, the measure of angle DFE is 59 degrees and option B is the correct choice.